# Limit with a big exponentiation tower

Find the value of the following limit:

I don’t even know how to start with. (this problem was shared in Brilliant.org)

Some of the ideas I tried is to take the natural log of this expression and reduce it to $\ln(a/b)$ then use L’Hopital’s but that made it false!!

I know the value of the limit it is $e^{-a}$ but please how to prove it?

Let’s observe that the limit is of form

where $f(x)=e^{e^{e^x}}$ and $\alpha=e^{-a}$. Since $f$ is differentiable, we have

for some $0\leq \xi \leq \frac{\alpha}{f'(x)}$. It remains to prove that $f$ is “continuous enough” for this limit to converge as it seems it should, that is

I’ll try to find some clever way to do this.

Edit: Well, brutal force will do. We have

Now,

Obviously, $e^\xi\to 1$. Furthermore, by above inequality,

and

That finishes the proof.